Equilibrium and Stability Properties of Self-Organized Electron ...

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PSFC/JA-01-10 Equilibrium and Stability Properties of Self-Organized Electron Spiral Toroids C. Chen, R. Pakter and D. C. Seward* April 2001 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139, USA *Electron Power Systems, Inc. 42 Washington Drive Acton, MA 01720, USA This work was supported in part by Ballistic Missile Defense Organization, Contract No. DSWA01-97-M-0537. Reproduction, translation, publication, use and disposal, in whole or part, by or for the United States government is permitted. Submitted for publication in Physics of Plasmas.

Transcript of Equilibrium and Stability Properties of Self-Organized Electron ...

Page 1: Equilibrium and Stability Properties of Self-Organized Electron ...

PSFC/JA-01-10

Equilibrium and Stability Properties of Self-Organized Electron Spiral Toroids

C. Chen, R. Pakter and D. C. Seward*

April 2001

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge, MA 02139, USA

*Electron Power Systems, Inc. 42 Washington Drive

Acton, MA 01720, USA This work was supported in part by Ballistic Missile Defense Organization, Contract No. DSWA01-97-M-0537. Reproduction, translation, publication, use and disposal, in whole or part, by or for the United States government is permitted. Submitted for publication in Physics of Plasmas.

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Ca-;0/1Rk=<-"-am02:<,1GHB0B<B<J.Ca-"31:<LW8-@ML;/.UF"L;Cm 0=<-35_*GM:<,5:<,.- ` E?S B2GMl"-NX+B2L:<, 0::<,.--@H-"F:<=<L;/.B

02=<-F"LK/.B2GM3.-"=<-"3D:<LRkLK=<C&0,.LK@H@ML?_oWT-N02C :<LK=<J.BTW p:<,.--"@M-"F":<=<LK/D:<-"CamT-"=?02:<J.=<-GHBa02B2B<J.Ca-3D:<LWT-@HL?_XTB<L:<, 02:0F"[email protected]"B<F"=<GMm.:<GHLK/rGMBJ.B<-"38F;p:<,.-CaGH/.LK==?03.GMJ.BLKR:<,.- ` EKS'GHBa02B2B<J.Ca-3r:<L

WT-aB<C0@M@F"LKCam 02=<-"3r_*GM:<,r:<,.-CD0?wkLK==?03.GMJ.BKXTB<L:<, 0::<,.-:<LK=<L;GM3 0@-8-F":<B02=<-/.-"UK@HGMU;GMW.@HVB<CD02@H@02/.353 p*:<,.-GHLK/1W 0F29NUK=<L;J./.31GHB0B<B<J.Ca-35:<LW8-u.vN-3 )S*,.-@A0B<:0B<B<J.Cam.:<GML;/1GHCm.@HGM-"B:<, 02::<,.-

-"y;JTGM@HGMW.=<GHJ.C:<,.-"LK=<VD0/.3B2J.W.B<-"y;J.-"/N:B2:?02W.GH@MGH:AVD0/ 02@HVNB<GMB02=<-d.02@HGM3GM/:<,.-:<GMCa-B<FN0@M-:<, 0:GHBB2,.LK=<:GH/F"LKCam 02=<GHB<LK/_*GM:<,:<,.-:<GMCa-B<FN02@H-RHL;=*:<,.-GHLK/CaLK:<GHLK/ )

S L:<,.-@ML?_-"B<:L;=<3.-=?X;_-:<=<-N02::<,.-:<LK=<L;GM3 [email protected]?_jGM/:<,.- ` E?S_*GH:<,r06B<CD02@H@0B<mT-"F":=?02:<GHLnkGQ) -N)MX

_*GH:<, [;X"_*,.-"=<- GHB :<,.-CD0?wkLK= ` E?S5=?023.GHJ.B?p 02BGMR.GH: h.L?_*B*0@MLK/TU0B2:<=?02GHUK,N: FVN@HGM/.3.-"=?XN0/.3

02/ 0@MVNl"-3.-":?02GH@M-"3D-"y;J.GH@MGHW.=<GHJ.C m.=<L;mT-"=<:<GM-"BLKR:<,.-m8LK@ML;GM302@ h.L?_'GH/:<,T- ` E?St02BB2,.L?_*/DGM/qITGMU8) [K)

PQ/:<,.-m8LK@HLKGH3 02@+F"LNLK=<3.GH/ 02:<-B2V;B2:<-Cn p<XN:<,.--"yNJ.GM@HGMWT=<GMJ.Ct-"@H-F":<=<LK/3.-"/.B<GH:AVGHB02B<B2J.C-"3:<LWT-

n p "!#$"!%&'%& Y( LK:<,.-"=<_*GHB<-N) n<[Kp

S*,.--"yNJ.GM@HGMWT=QGHJ.Ct-"@M-"F":<=<LK/h.L?_sdK-"@HL;F"GM:QV, 0B*:<,.-UK-"/.-"=?02@+RkLK=<C

)* n p *,+ n p )- +/.0*(1 )- 1 n<ZKp

_*,.-"=<- *,1 *(+ 02/T3 *,1 F"LK/.B<:?)S*,.-u.vN-3GML;/3.-"/.B<GH:AVGHB0B<B<J.Ca-"3:<LaWT-

32 n p54 n p6 n<\Kp

_*,.-"=<-74 GMB6:<,.-GML;/Rk=?02F":<GML;/ XN_*,.GMF,GHB*B<@HGHUK,N:<@MVU;=<-;0:<-=*:<,02/J./.GH:QV;)PQ:aGMB=<-;03.GM@HV5B<,.L?_*/q:<, 0:a:<,.--"yNJ.GM@HGHW.=<GMJ.C&-"@M-"F":<=QL;/q3.-"/.B<GH:AVs02/.3h.L?_tdK-"@HL;F"GH:AVNX 0/.3 )*

3.-"u./.-"3GH/ ` yNB?)+n<[Kp*0/.31n<Z;p<X;BK02:<GHB<RkV:<,.-B<:<-N023.VNgiB<:?0:<-FL;/;:<GH/NJ.GM:QV-"y;J 0:<GML;/

8 -:9 9#;

.=<?> )@ sY n b p

Page 7: Equilibrium and Stability Properties of Self-Organized Electron ...

_*GH:<, 9 9#; sYK) PQ/ ` y8)*n b p<X )@ 8 - )* GMB:<,.--@H-"F:<=<L;/aF"J.=<=<-"/;:3.-"/.B<GH:AVD02/.3 8 - GHB:<,.-6-"@M-"F":<=<LK/F2, 0=<UK-N)

S*,.-F"[email protected];=<F-W 0@A0/.F"--"yNJ 02:<GHLK/RkLK=*:<,.--"yNJ.GM@HGMW.=<GHJ.Ct-"@M-"F":<=QL;/h.L?_sGMB

9)*

9:;. n )* >< p )* 8 -

). )* n/ )- 1 . ) p " n<Kp

_*,.-"=<- 9 9:; Y;X GMB:<,.--"@M-"F:<=<L;/1CD02B<BKX/ )- 1GMB:<,.-r02m.mT@MGH-35:<LK=<LKGH3 0@CD02U;/.-":<GMFu.-"@M3X0/.3) n p )- 02/.3 ) + n p )- + . 1 n p )- 1D0=<-:<,.-B2-@HRkgi-"@M-"F":Q=<GHF50/.3B<-"@MRkgiCD02U;/.-:<GHFu.-"@H3.Bm.=<LN3.J.F"-3WNV:<,.--@H-"F:<=<L;/.B02/T3W 02F9;U;=<LKJ./.3GHLK/.BKXN=<-"B<m8-F":<GHdK-"@MVN) e @H:<,.LKJ.U;,502/50m.m.@MGH-"3:<LK=<LKGH3 02@

CD02UK/T-:<GHFu.-@H3GHBGH/.F"@MJ.3T-3GM/:Q,T-m.=<-B2-/N:RkLK=<CD02:<GHLK/LKR+:<,.--"yNJ.GM@HGHW.=<GMJ.Cz0/.3B<:?0W.GM@HGM:QVam.=<LKWT@M-"CX

GH:GHBB2-:*:<LWT-l"-"=QLGH/102@M@8L;R+:<,.-/;J.Ca-"=<GMFN02@8-"v+02Cam.@M-"B*m.=<-"B<-/N:<-"3@Q02:<-"=*GM/:<,.GMB*m 02m8-"=?) S*,.-=?023TGA02@

F"LKCamTL;/.-/N:*L;R ` y+)+n<Kp GHB

* + n p -

n p .0* +Nn p / . 1 n p n Kp

_*,.-"=<-:<,.-/.-"UK@HGMU;GMW.@HVB<CD02@H@8:<-"=<C * 1 + , 0B*W8--"/GHUK/.L;=<-3)*E?LK@HdNGM/.U ` y+)+n Kp RkLK= * +Nn p [email protected]

* +Nn p [Z

- n/ . 1 p -

n/ . 1 p . b -

!

n<fKp

=<-"B<mT-"F":<GMd;-@HVN)S*,.--"yNJ.GM@HGHW.=<GMJ.C O 02vN_-"@M@+-"y;J 0:<GML;/.B0=<-;|

[ 9

9 n p 8 - n<[ 8 4 p n p6 n<Kp

99 1 # - * +Nn p n p n<]Kp

E?LK@HdNGM/.U ` y+)+n<Kp RkL;= _*GM:<,:<,.-WTL;J./.3 02=<VF"LK/.3.GH:<GML;/ n7jY;pjYVNGM-"@H3.B

n p 8 n - p<n<[ 8 4 p "! n<[ 8 "! p6$"! %&'% Y, '%&"! n<[;YKp

E?J.W.B2:QGH:<J.:<GH/.U ` y8)+n<fKpGM/N:<LDn<]Kp VNGM-"@H3.B

1 -

Z 2n p - n/ . 1 p -

n/ . 1 p . b -

!

n<[;[Kp

Page 8: Equilibrium and Stability Properties of Self-Organized Electron ...

_*,.GHF2,, 02B6:<LW8-B<LK@HdK-"3/NJ.Ca-=<GHF;0@M@HVJ./.3.-"=*:<,.-W8LKJ./.3 0=<VFL;/.3.GM:<GHLK/ 1 n7 psYK)

xB<-aGHBCD023.-aL;R0aFL;Cm.J.:<-"=FLN3.-NX+/ 0C-"3j(*ITS ` E?SX8:<LB<L;@MdK- ` yNB?)n<f;p0/.3jn<[K[Kp*/NJ.Ca-=<GHg

FN02@H@MVN)rIKLK=m.=<-"B<-"/N:mTJ.=<mTLKB2-BKX+GH:GMBF"LK/NdK-"/.GH-/N::<L-"vNm.=<-"B<B ` yNB?)n<fKp02/.3jn<[;[Kp*GH/1:<,.-aRkL;@M@HL?_*GH/.U

3.GHC-"/.B<GHLK/.@H-"B<B*RkL;=<CBnkRkLK=[ "! p

n p [Z . n . p . b n64 8 [;p [

[ 8 [

!

n<[;ZKp

02/.3 Z . n . p . b n64 8 [Kp [

[ 8 [

!

n<[;\Kp

B<J.W+wk-"F:6:<L:<,.-W8LKJ./.3 0=<VFL;/.3.GH:<GMLK/

"! sY n<[ b p

PQ/ ` yNB?)+n<[KZ;p*02/.35n<[K\Kp<XN:<,.-3.GHCa-/.B2GMLK/T@M-"B<B6d.02=<GA0W.@H-B02/.3m 02=?02Ca-":<-=<B02=<-3.-"u./.-"3WNV

7 "!

n "! p *(+

n "! p

n "! p - 1

n "! p n<[;Kp

n "! p -

n "! p

"!

n"! p

_*,.-"=<-

n "! p -

! n<[ Kp

GHB*:<,.--"@M-"F":<=<LK/m.@Q02B2C0Rk=<-yNJ.-"/.F"Vr02: 7 "! X+02/.3 GMB6:<,.-B<m8--"3L;R @MGHUK,N:*GH/ ;)` yNB?)+nQ[;ZKp*02/.31n<[K\;p0=<-RkJ.@H@MVF2, 02=?0F:<-"=<GHl-"3W;V:<,.=<--3.GMCa-"/.B<GHLK/.@H-B2Bm02=?02Ca-":<-=<B?XN/ 02Ca-"@MVNX

X

03.GMCa-"/.B<GHLK/.@H-B2BCa-N02B<J.=<-*L;RT:<,.-*-"@M-"F":<=<LK/a3.-/TB<GM:QVNX 4 X?:<,.-*=?0:<GMLLKR8:<,.-*W 02F9;U;=<LKJ./.3aGML;/3.-"/.B<GH:AV:<L:<,.-6-"@M-"F:<=<L;/a3.-"/.B<GH:AVNXT0/.3 XK:<,.-6/.LK=<CD02@HGMl"-"3aF"VNF@HLK:<=<LK/Rk=<-yNJ.-"/.F"V0B<B<LNF"GA0:<-3_*GM:<,:<,.-02m.m.@HGM-"3:<LK=<L;GM3 [email protected];/.-:<GHF6u.-@H3 ) O L;=<-L?d;-=?XKB2GM/.F"-6:<,.-*d.0=<GA0W.@M- d.0=<GM-"BL?dK-"=:<,.-6=?02/.U;-*Rk=<L;C [*:<L "!

f

Page 9: Equilibrium and Stability Properties of Self-Organized Electron ...

02/.3D:<,[email protected];R "! FN02/DWT-F,.LKB<-"/50=<W.GM:<=?0=<GM@HV;XT_--vNmT-"F":6:<, 0: ` E?S'-"y;J.GH@MGHW.=<GQ0-"vNGMB2:6L?dK-"=

0_*GM3T-=?0/.UK-L;R B<VNB<:<-"Ctm 02=?02Ca-":<-=<BK)

ITGHUKJ.=<-ZB<,.L?_*B0:<-"/NJ.LKJ.B ` E?S'-"yNJ.GH@MGHW.=<GMJ.C&0BLKWT:?02GM/T-3r/;J.Ca-"=<GMFN02@H@MV_6GM:<,D:<,T-0GM3rLKR:<,.-

F"LKCam.J.:<-"=6F"LN3.-GM/D:<,.-0W.B<-"/.F"-LKR02/NV10m.m.@MGH-"3:<L;=<LKGH3 02@ CD02U;/.-:<GHFu.-"@M3)*S*,.-F,.LKGHF-LKRB2V;B2:<-C

m 02=?0C-":<-"=<BGH/jITGHUT)ZaFL;=<=<-B<m8LK/.3.B:<LT|

[ ;X40 [ YKZ;X YKX*0/.3 "! [ Y;[K)PQ/:<,.GHB6F;0B<-NXN:<,.- ` E?S'-yNJ.GH@MGHW.=<GMJ.C GHB6:<-/NJ.L;J.B6WT-"FN02J.B<-:<,.-+J.3.9;-=6m 0=?02Ca-":Q-"=6GHB6-d.0@MJ02:<-"3:<LaWT-

n

Z;p<n "! 8 [;p [KY X_*,.GHF2,qGHBaB<CD02@H@*F"LKCam 0=<-3q_6GM:<,qJ./.GH:QV;)qPA/oI.GHUT)6ZKX:<,.-

/.LK=<CD02@HGHl-"3FVNF"@ML;:<=<LK/Rk=<-yNJ.-"/.F"V n"! p02/.3/.L;=QCD0@MGHl-"3-@H-"F:<=<L;/'0/.UKJ.@Q02=h.L?_ dK-"@HL;F"GH:AV

n"! p602=<-m.@HLK:<:<-"3502B0RkJ./.F":<GML;/L;R :<,.-/.LK=<CD02@HGMl"-"3CaGH/.LK=6=?023.GHJ.B #"! RkLK=6:<,.-F"@MLNF29N_*GHB<-"g=<LK:?0:<GM/.U-"@M-"F":<=<LK/Dh.LK_'GH/D:<,.-:<-"/;JTLKJ.B ` E?S'-"yNJ.GH@MGHW.=<GMJ.CD)aIKL;=:<,.-F"@MLNF29N_*GHB<-"gi=<LK:?0:<GM/.U-"@M-"F":<=<LK/

h.L?_X._*,.GHF2,FL;=<=<-B<m8LK/.3TB*:<La:<,.-CGH/NJ.B6B<GMU;/GM/ ` yNB?)+n<[;ZKp602/.35n<[K\Kp<X.:<,.--"@M-"F":<=<LK/.B6=<LK:?0:<-GH/:<,.-

mTL;B<GH:<GMd;- )- +?gi3.GM=<-"F":<GML;/1GH/5=<-Rk-"=<-"/.F-:<LD:<,.-B<F,.-CD02:<GHF3.GQ02U;=?02CB2,.L?_*/5GM/sITGMU8)[KX602/.35:<,.-B2-@HRkg

CD02UK/T-:<GHF6u.-@H3am8LKGH/;:<BLKJ.:L;R8:<,.-6m 02U;-*W8-"F;0J.B<-6:<,.-6FJT=Q=<-"/N:h.L?_GMBGM/a:<,.-6/.-"UT0:<GMd;- )- +"gi3TGM=<-"F:<GHLK/ )PQ/FL;/;:<=?0B<:?XW8LK:<,:<,.-D-@H-"F:<=<L;/=<LK:?02:<GHLK/o0/.3B<-"@MRkgAC0UK/.-":<GMFDu.-"@M3RkL;=:<,.-F"LKJ./N:<-"=<giF"@MLNF29N_*GHB<-"g=<LK:?0:<GM/.U-@H-"F:<=<L;/h.L?_=<-"dK-"=<B<-:Q,T-GH=3.GM=<-"F":<GML;/.B_*GH:<,=<-B2mT-"F::<L:<,.-6F"@HL;F9;_6GMB<-"gi=<L;:?02:<GH/.U-"@M-"F":<=<LK/

h.L?_)

cGH:<,D:<,.-F2,.L;GMF"-L;R60/DGM/./.-"=CGH/.LK==?023.GHJ.BLKR "! b ]; [KY C&02/.3q0CD0?wkLK==?023.GHJ.BLKR

jZ [KY CRkL;=:<,.- ` E?SX;_-3.-"=<GMd;-Rk=<LKCzITGHUT)8Z6:<,.-RkL;@M@HL?_*GH/.U3.GMCa-"/.B<GML;/ 02@8m02=?02Ca-":<-=<B?| j Y;Y [KY CDX *(+ n ps \Kf [;Y C2B?X 1 n "! p 8 [KY S -"B<@Q0KX sZ Z b \ [;Y ! X02/.3 2 'Z ZK; [KY ! X._*,.-"=<- GHB6:<,.-:<LK:?0@+/NJ.CW8-=6L;R:<,.--"@H-F":<=<LK/.BGM/:<,.- ` E?St02/.3 2 GHB:<,.-:<LK:?0@+/;J.CWT-"=*LKR:<,.-GMLK/TB*GH/:<,.-W 02F9NUK=<LKJ./.3)

ITGHUKJ.=<-D\B<,.L?_*Bm.@ML;:<BL;R:<,.-a/.LK=<CD02@HGMl"-"3rF"VNF@HLK:<=<L;/rRk=<-"y;J.-"/.F"V n"! p02/T3r/.LK=<CD02@HGHl-"3-"@M-"F":<=<LK/j02/.U;J.@A0=hTL?_ dK-"@MLNF"GM:QV n"! p*dK-"=<B<J.B:<,.-/.LK=<CD02@HGMl"-"31CGH/.LK==?023.GHJ.B#"! RkL;=:<,.-F"@MLNF29N_*GHB<-"gi=<LK:?0:<GM/.U-"@H-F":<=<LK/h.L?_5GH/0/GM/N:<-/TB<- ` E?S5-"yNJ.GM@HGMWT=<GMJ.Ct0BLKW.:?0GM/.-"3/;JTC-"=<GMFN02@H@HVGM/:<,.-

02W.B2-/.F"-L;R+02/NVa02m.m.@HGH-3:<LK=<LKGH3 02@;CD02U;/.-:<GHFu.-"@M3RHL;=+:<,.-F2,TLKGMF"-L;RNB<VNB<:<-"Cjm02=?02Ca-":<-=<B FL;=<=<-B2mTL;/.3.g

GH/.U:<L8|

s[KY;YKYKX 4 s[ YKY;YK[;X sYKX+02/T3 "! s[ Y;[ PQ/:<,.GHB*FN02B2-;XN:<,.- ` E?Sj-"yNJ.GH@MGHW.=<GMJ.C

Page 10: Equilibrium and Stability Properties of Self-Organized Electron ...

GHBaGM/N:<-"/.B<-W8-FN0J.B<-:<,.-5+J.3T9K-"=m 0=?02Ca-":<-=aGMB-d.0@MJ 0:<-3q:<LrW8- n

ZKp<n "! 8 [;p KX

_*,.GHF2,1GHB@Q02=<U;-=:<, 02/1J./.GH:QV;)PQ/1F"LKCam 02=<GHB<LK/5_*GH:<,1:<,.-:<-/NJ.L;J.BFN02B<-B<,TL?_*/1GH/jITGMU8)Z;X _-L;W.g

B<-"=<dK-DGM/oITGMU8)6\:<, 02::<,.-DC0UK/.GH:<J.3.-L;R:<,.-D-@H-F":<=<LK/h.L?_ d;-@HLNFGH:QV5GM/TF=<-N02B2-B=?02mTGM3.@HVs0:a:<,.-

LKJ.:<-"=-"3.UK-LKR:<,.- ` E?S).S6,.GMB:<=<-"/.3DW8-F"LKCa-"BCL;=<-m.=<L;/.LKJ./.F"-"3q02B:<,.-+J.3T9K-"=6m02=?02Ca-":<-=GH/.g

F"=<-;0B<-"B?)qcGM:<,q:<,T-F,.LKGHF-L;Ra02/qGH/./.-"=aCGH/.LK=a=?03.GMJ.BLKR "! [ Y;Y [KY C 02/.3o0CD0?wkLK==?023.GHJ.BL;R &Y [C RkLK=:<,.- ` E?SX_-D3.-=<GHdK-DRk=<LKC ITGMU8)6\:<,.-RkL;@M@HL?_*GM/TU13.GHC-"/.B<GHLK/ 0@6m 0g

=?02Ca-":<-=<BK| [ YK[ [;Y CDX *,+ n p ] b ] [KY C2B?X 1 n "! p 8 [ b ] [KY S-B2@A0KX s KY;]K]K;f [KY ! XN0/.3 2 s K[;YK b ] [KY ! X?_*,.-"=<- TGHB :<,.-:<L;:"0@N/;JTCWT-"= L;R.:<,.--@H-"F:<=<L;/.BGH/:<,.- ` E?S02/.3 2 GMB6:<,.-:<LK:?0@8/NJ.CWT-"=*LKR:Q,T-GMLK/TB*GH/:<,.-W 02F9NUK=<LKJ./.3)

]

Page 11: Equilibrium and Stability Properties of Self-Organized Electron ...

"? *;1 PQ/:<,.-m.=<-"B<-"/N:02/02@MVNB<GHB?X._-F"LK/.B2GM3.-"=6-"vN:<=?02LK=<3TGM/ 0=<V;gACLN3.-m8-=<:<J.=<W 0:<GML;/.B6L;/:<,.-F"[email protected]

` E?S-yNJ.GH@MGHW.=<GMJTCo3.GHB<F"J.B<B<-"3aGH/DE?-"FN)?P<P").x/.3.-"=:<,.-6-"v;:<=?0LK=<3.GH/ 02=<VNgiCaL;3T-*m8-=<:<J.=<W02:<GML;/.B?XK:<,.-6-"@M-"F"g

:<=<LK/j3.-"/.B<GH:AV 2n# ; p<X*-"@M-"F:<=<L;/h.L?_ d;-@HLNFGH:AV )* n# ; p<X02/T3-"@M-"F":<=<GMFq0/.3CD02U;/.-":<GMFDu.-"@H3.B) n# ; p602/.3

) n# ; p GH/:<,.- ` E?SjFN02/WT--"v;m.=<-"B<B2-3502B

2n# ; p 2n p . 2n p - 2 + n<[;fKp

)* n# ; p * +Nn p )- + . * n p )- . * +Nn p )- + - 2 + n<[;Kp

) n ; p n p )- . n p )- . +Nn p )- + - 2 + n<[;]Kp

) n# ; p 1?n p )- 1 . 1"n p )- 1 - 2 + n<Z;YKp

PQ/ ` y;BK)*n<[KfKpkgn<ZKYKp<X7 [( Z( >>> X GHB:<,T-Rk=<-yNJ.-"/.F"V1L;R6:<,.-m8-"=<:<J.=<W 02:<GHLK/.B?X GMBa:<,.-5nkCGHg

/.LK=?p=?03.GA0@6F"L;L;=<3.GM/02:<-NX GHB:<,.-DmTLK@HLKGH3 02@a02/.U;@M-NX02/.3 GHB:<,.-:<L;=<LKGH3 02@a02/.UK@H- *S6,.-yNJ 02/N:<GHg:<GH-B n p<X *(+ n p )- + X n p )- 0/.3 1 n p )- 1 0=<-:<,.--"@M-"F:<=<L;/53.-"/.B<GH:QV;XhTL?_d;-@HL;F"GH:AVNX60/.35-"@M-"F"g

:<=<GMFD02/.3DCD02U;/.-":<GMFu.-"@H3.BGH/D:<,.-F"[email protected] ` E?S'-"yNJ.GM@HGMWT=<GMJ.CDXT=<-"B<m8-F":<GMd;-"@MV8S*,.-m8-=<:<J.=<W 0:<GML;/.B n# ; p n p - 2 + X *,+ n# ; p * n p )- . *,+ n p )- + - 2 + X;-":<FN)802=<-02B<B<JTC-"3:<LWT-B<CD02@H@+GH/:<,.-m.=<-"B<-/N:0/ 02@HV;B2GMB?)+-"FN02J.B<-02/-"GHUK-"/.CLN3.-_*GH:<,PAC YGMB "X._-CJ.B<:

B<,.L?_:<, 0:02@H@T-"v;:<=?0LK=<3.GH/ 02=<VaCLN3.-"B, 0<d;-6PQC Y*:<Lm.=<L?d;-6:<, 0:0/ ` E?S-"yNJ.GM@HGMW.=<GHJ.C GHBB<:?0W.@M-02U802GH/.B<:*-"vN:<=?02LK=<3.GH/ 0=QVNgiCaLN3.-m.J.=<:<J.=<W 0:<GMLK/TB?)

S*,.-02GMCaB6LKR :<,.GHB6B<-"F:<GHLK/502=<-:<La3.-=<GHdK-0F"LKCam.@H-:<-B2-:6LKR @HGM/.-N02=<GHl"-3-yNJ 0:<GML;/.B*=<-"@A0:<GM/.Ua:<,.-

mT-"=<:<J.=<W8-3ayNJ 02/N:<GM:<GH-"B 2X * X * +NX X +NXT0/.3 1"XK:<L3.-"=<GMd;-*:<,.-*-"GHUK-"/Nd.02@MJT-*-"y;J 0:<GML;/0/.302m.mT=<LKm.=<GQ02:<-6W8LKJ./T3 02=<VaFL;/.3.GM:<GHLK/.BKXT0/.3:<LGH/;d;-"B<:<GMU802:<-:<,.-6B2:"0W.GH@MGH:AVam.=<LKm8-=<:<GH-"BLKR+WTL;:<,:<-/NJ.L;J.B02/.3GM/N:<-/TB<- ` E?S*B?)( LK/.B2GMB<:<-"/N:+_*GM:<,:<,.-:<,.GH/.giWT-N0C 0m.m.=<L?v;GHCD02:<GHLK/ X":<,.-m.=<-"B<-"/N:02/ 02@HVNB<GHB+GMU;/.LK=<-"B

F"J.=<d.02:<J.=<-D-T-"F":<BGH/:<,T-D:<LK=<LKGH3 [email protected]=<-"F:<GHLK/:<, 02:GMB?X*:<,.-DJ./TGM:d;-"F:<L;= )- 1 GHBFL;/.B<GH3.-=<-"3 02B10

F"LK/.B<:?0/;:J./TGM:dK-"F":<LK=?) ^ -":[email protected]"=<GMd.0:<GML;/.Ba02=<-m.=<-"B<-"/;:<-"3DGM/ e m.m8-/.3.GHF"-B e g ^ RkL;=a02/ ` E?S'GH/

LKm8-"/B2m 02F"-nk-N) UT)MX?GM/0GM=?p<XN0/.3:<,.-=<-"B<J.@H:<B*02=<-B<J.CaC0=<GMl"-"3GH/E?-"F;)"P<P<P) e )"S*,.-B<:?02W.GH@MGH:AVm.=<LKm8-"=<:<GM-"B

LKR:<-/NJ.LKJTB02/T3GM/N:<-"/.B<- ` E?S6B02=<-3.GHB<F"J.B<B2-3GM/1EK-FN)NP<P<P)H)

[KY

Page 12: Equilibrium and Stability Properties of Self-Organized Electron ...

* S*,.-DB<:?02W.GH@HGM:QV5m.=<LKWT@M-"C&LKR02/ ` E?S FN02/:<,.-/WT-DRkLK=<CJ.@Q02:<-"3o02B:<,.-RkLK@H@ML?_6GM/.U1-"GMU;-"/;[email protected]

m.=<LKWT@M-"CnkRkLK= "! %&'% p[

[ 8 M[ . n p 8

H[ . + n p sY( n<Z;[Kp

_*GH:<,:<,.-W8LKJ./.3 0=<VFL;/.3.GM:<GHLK/.BK|

?n7 "! p Z! "! @ n "! p@#" n "! p n<Z;ZKp

02/.3

n7 p Z! $ n p$ " n p n<Z;\Kp

RkLK=02/ ` E?SGH/L;mT-"/B2m 02F"-ank-;) UT)HX;GM/102GH=?p<) PA/ ` yNB?)8n<ZK[;pkg2n<ZK\;p<X;:<,.--GHUK-"/.RkJ./.F":<GML;/ #rGHB3T-u./.-"3W;V

6n p +Nn p n<Z b p

S*,.-B<JTB<F"-m.:<GHW.GM@HGH:AVRkJ./.F":<GML;/.B n p*02/.3 + n p*0=<-3.-u./T-3W;V

n p 8 n<[ 8 p [ 8 n<Z;Kp

+Nn p 8 [ .

[ 8 . Z

8 Z

[ 8 . [ 8 n 8 2p<n<[ 8 p

n 8 Z p n<Z Kp

=<-"B<mT-"F":<GMd;-@HVNX;_6,.-=<- . ) S*,.-RkJ./.F":<GML;/ n p GHB*3.-"u./.-"3WNV

n pjn 8 2p [ .

[ 8 . n 8 Z 2p [ n

p 8 n<Z;fKp

S*,.-03.CGH:<:?02/.F"-RkJ./.F":<GML;/% n p GMB63.-"u./.-"3WNV

n p'&)( +Nn# ; p 1 n# ; p * Z! +Nn# ; p *

[Z!

1 n p + n p n<Z;Kp

_*,.GHF2,F;0/WT--"vNm.=<-"B<B<-"3-"v;mT@MGHFGH:<@MVr0B

n p Z! [

[ 8

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Page 13: Equilibrium and Stability Properties of Self-Organized Electron ...

8

[ n 8 Z p .

8

n 8 p [ 8

[ 8 [

n<Z;]Kp

S*,.-RkJ./.F":<GHLK/.B @ n p02/.3 $ n pa02=<-:<,.-u.=<B<:<gA9;GH/.3' -"B<B<-"@02/.3o~60/.9K-"@6RHJT/.F:<GHLK/.BLKRL;=<3.-"= X

=<-"B<mT-"F":<GMd;-@HVN)e @M:<,.L;J.UK,r:Q,T--"GHUK-"/;dT02@MJT--"yNJ 02:<GHLK/nQZ;[Kp=<-"B<-"CW.@H-"B:<,.--"v;:<=?0LK=<3.GH/ 02=<V-"GMU;-/[email protected]"yNJ 02:<GHLK/

H[KKXH[ 3.-"B<F"=<GMWTGM/.Ur:<,.--"@H-F":<=<LK/5h.L?_tGM/5F"=<LKB2B<-"3.giu.-"@M35d.0FJ.JTC-"@M-"F":<=<LK/53.-"dNGMF"-"BB<J.F,s02BCD02U;g

/.-":<=<LK/.B60/.3F=<L;B<B<-"3.giu.-"@[email protected]"=<B?X?:<,.-"=<-0=<-B<GHUK/.GHu.FN02/N:3.G 8-=<-"/.F"-"B?)ITGM=<B<:?X?:<,T-m.=<-"B<-"/N: -"GHUK-"/[email protected] 0:<GML;/1GM/.F"@HJ.3.-B:<,.--8-F":<BL;R02/1u.vN-"31GML;/1W 02F9;U;=<LKJ./.31_6,.GMF,1GMB02W.B<-"/N:GM/1F"=<LKB2B<-"3.gu.-"@M3jd.0FJ.JTC -"@M-"F":<=<LK/j3.-"d;GHF"-BK)oS*,.GMB502@M@HL?_*BJ.B:<Ls02/ 0@MVNl"-D:<,.-rB<:?02W.GH@HGM:QVqm.=<L;mT-"=<:<GM-"BL;R:<,.-

B<-"@MRkgALK=<UT0/.GMl"-"3 ` E?SjGM/:<,.-a02W.B2-/.F"-LKR*02/r0m.m.@MGH-"3:<LK=<L;GM3 0@8C0UK/.-":<GMFu.-"@H3 )*E?-F"LK/.3X;:<,.-m.=<-B2-/N:

-"GMd;-/[email protected]*-"yNJ 02:<GHLK/CJ.B<:W8-*B<L;@MdK-"3aB<J.W+wk-F"::<L:<,.-*L;mT-"/W8LKJ./.3 0=<VFL;/.3.GH:<GMLK/aUKGHdK-"/GH/ ` y8).n<Z;\Kp

=?02:<,.-"=:<, 02/D0*F@HLKB2-3aF"LK/.3.JTF:<GH/.UKgA_0@M@TWTL;J./.3 02=<VF"LK/.3.GH:<GML;/aJ.B<-"3aGM/a:<,.-02/ 0@MVNB<GHBLKR8F"=<LKB<B2-3.gAu.-@H3

d.02F"J.J.Ct-"@M-"F":<=<LK/3T-dNGHF-"B?)PQ/D:<,.--"GHUK-"/;[email protected]"yNJ 02:<GHLK/n<ZK[Kp<X8:<,.-"=<-02=<-:Q_LGHCm8LK=<:?02/N:=<-B<L;/ 02/.F"-"BRkL;=:<,.-GH/;:<-"=?02F":<GHLK/

WT-":Q_--"/:<,.--"@H-F":<=<LK/h.L?_ 0/.3:<,.-mT-"=<:<J.=<W8-3_0<dK-u.-"@M3.BKX;/ 0C-"@HV;X

8 2n p Y, n<\;YKp

02/.3

n psn 8 2p [ .

[ 8 . n 8 Z 2p [ n

2p 8 Y n<\;[Kp

e FL;Cam.J.:<-=+F"LN3.-NX/ 0C-"3(*ITS ` E?SX, 02B+W8-"-/3T-d;-@HLKm8-3RHL;=+B<LK@HdNGM/.U6:<,.--GHUK-"/[email protected].=<LKW.@H-C

3.-"u./.-"3rGH/ ` yNB?)n<Z;[Kpkg2n<Z;\Kp<)aPA/1:<,.-aF"L;3.-NX+:<,.-aB<:?0/.3 02=<31B<,.LNLK:<GH/.UC-":<,.LN3rGHBJ.B2-31:<LGM/N:<-"UK=?02:<-

:<,.--"GMU;-"/;[email protected] 02:<GHLK/qn<ZK[;pRk=<L;C "! :<L7 X 0/.3:<,.-WTLKJ./T3 02=<VF"LK/.3TGM:<GHLK/.BGM/ ` yNB?) n<Z;ZKp

02/.35n<ZK\;p<) S*,.-F"L;3T-,02B*W8-"-/d.02@HGM3 0:<-3502/.3GHB6_L;=<9;GH/.UaFL;=<=<-F":<@HV;)

xB<-GMBCD023.-LKR(*ITS ` EKS:<LGM/Nd;-B<:<GHUT0:<-6:<,.-B<:?02WTGM@HGM:QVm.=<L;mT-"=<:<GM-"BL;R8:<-"/NJ.LKJ.B ` E?S-"y;J.GH@MGHW.=<GQ0

GH/q:<,.-502W.B2-/.F"-L;Ra02/NVs02m.m.@HGH-3:<LK=<LKGH3 02@6CD02U;/.-:<GHFu.-"@M3 )oI?L;=a:<,.-:<-"/NJ.LKJ.B ` E?St-"yNJ.GH@MGHW.=<GMJ.C

[KZ

Page 14: Equilibrium and Stability Properties of Self-Organized Electron ...

B<,.L?_6/5GM/'I.GHUT)*ZaGH/sE?-"FN)+P<P"X :<,.-=<-"B<J.@H:<BL;R6:<,.-B<:?02W.GH@MGH:AVj0/ 02@HV;B2GMBD02=<-B<J.CaC0=<GMl"-"35GM/'I.GHUT) b )

ITGMU;J.=<-"B b n<0Kp02/.3 b nkW p6B<,.L?_X=<-B<m8-"F:<GHdK-"@MVNX:Q,T-/.L;=QCD0@MGHl-"35=<-;0@-"GMU;-/TRH=<-"yNJ.-"/.FVj4- n "! p02/.31/TLK=<CD02@MGHl"-31GHCD02UKGH/ 02=<Vr-"GMU;-/.Rk=<-"yNJ.-/TFVrPAC n"! p0B0RkJ./.F":<GML;/1LKR*:<,.-mTLK@HLKGH3 [email protected]

/NJ.CW8-= RkLK=:<,.--"v;:<=?0LK=<3.GH/ 02=<V5-"GHUK-"/.CLN3.-m8-"=<:QJT=<W 02:<GHLK/.B?) e BB<,.L?_*/GM/oITGMU8) b XRkLK=-;0F2,

mTL;@ML;GM3 0@ CLN3.-/NJ.CW8-= Rk=<LKC o[:<L 02/.3DRkLK= [KY;X.:<,.-"=<-GMBa0B2:?02W.@H--GHUK-"/.CaL;3.-_6GM:<,q0=<-N02@-"GMUK-"/.Rk=<-"yNJ.-/.F"VN)a~LK_-"dK-"=?XTRkL;=-N02F2,rmTLK@HLKGH3 [email protected]/;J.CWT-"= Rk=<LKC f:<Lr];XT:<,.-"=<-GMB0

m 02GH=L;R+-"GMU;-/TCLN3.-"B_6GM:<,FL;CmT@M-"va-"GMU;-/.Rk=<-"yNJ.-/TFGH-BK) S*,.--"GMU;-/.CaLN3.-"B_6GM:<,mTLKB2GM:<GHdK-GHC0UKGH/ 02=<V

-"GMU;-/.Rk=<-"yNJ.-/TFGH-B02=<-J./.B<:?02WT@M-NX+_*,T-=<-N02B:<,.L;B<-_*GH:Q,5/.-U802:<GHdK-GMCD02U;GM/ 0=<VD-"GHUK-"/.Rk=<-yNJ.-"/.F"GM-"B02=<-

3 02Cam8-3 )S*,.-C0v;GHCJ.CtUK=<L?_6:Q,=?02:<-LKR :<,.-J./.B<:?0W.@M--"GMUK-"/.CaLN3.-B6GMB*PQC Y Y;YK n"! p<X_*,.GHF2,5L;F"F"J.=<B02: K)S*,.-=<-;0@-GHUK-"/.Rk=<-"y;J.-"/.F"VrGH/.F=<-N02B2-B02B:<,.-m8LK@HLKGH3 02@CaL;3T-/;JTCWT-"=

GH/.F=<-N02B2-BKX80B*-"vNmT-"F:<-"3Rk=<L;Ct:<,.-=<-B<L;/ 02/.F"-F"LK/.3.GH:<GML;/.B*GH/ ` yNB?)+n<\;YKp*0/.31n<\K[;p<)

e B:<,.-D-@H-F":<=<LK/3T-/.B2GM:QV5GH/.F"=<-;0B<-"BGM/:<,.- ` E?SXGQ) -N)MX0B:<,.-Dm 02=?0C-":<-=GM/.F"=<-N02B<-"B?X:<,.-

UK=<L?_6:<,D=?0:<-BLKR:<,.-GH/.B<:?02W.GH@HGM:<GH-BB<,.L?_6/DGH/qITGHUT) b 3.-"F"=<-;0B<-=<-"@A0:<GMd;-:<L:<,.-m.@Q02B<CD0Rk=<-yNJ.-"/.F"V;)

S*,.GHBGMBGH@[email protected]<:<=?02:<-"35GM/sITGMU8)KX GH/1_6,.GMF,1:<,.-/.LK=<CD02@HGMl"-35GHC0UKGH/ 02=<Vr-"GHUK-"/.Rk=<-yNJ.-"/.F"VrGMBm.@HLK:<:<-"302B0RkJ./.F":<GML;/1LKR*:<,.-m 0=?02Ca-:<-"=

RkL;=B<-"dK-"=?02@J./.B<:?02WT@M--"GMUK-"/.CaLN3.-B_*GM:<,5:<,.-am8LK@HLKGH3 [email protected]

/NJ.CW8-= sKX ;XTf02/.3D;)+S6,.-6F2,.LKGHF"-6LKR8B<VNB<:<-"C m 02=?0C-":<-"=<BGH/DITGHUT)TF"L;=Q=<-"B<m8LK/.3.B:<LT| sYKX

"!t[ YK[;X02/.3 4 8 [ Y Y;\KZ()PA/jITGMU8)KX :Q,[email protected];R:<,.-m 02=?02Ca-":<-= 4 GHB023.wkJ.B<:<-"3

02F"F"LK=<3.GH/.UD:<Lr:<,.-=<-@Q02:<GHLK/54 8 [ zY Y;\KZ,XB<Lr:<, 02::<,.-/.L;=QCD0@MGHl-"35-"@M-"F":<=<LK/5m8LK@HLKGH3 02@*h.L?_

dK-"@HL;F"GH:AVGHBu.vN-3q0: "! n "! p6 \ Z [KY B<-"- ` y+) n<fKp Q)PA:GHB-"d;GH3.-"/;:GH/qITGMU8) :<, 02::<,.-

/.LK=<CD02@HGHl-"3U;=QLK_*:<,=?02:<-*PQC n "! pT3.-"F=<-N02B2-B02B GM/.F"=<-N02B<-N) e B60C0:<:<-=L;RTR<02F":?X?:<,.-*=<-@Q02:<GHdK-CD02vNGMCJ.C'U;=<L?_*:<,=?02:<-*LKRT:<,.-*GM/.B2:?02W.GH@MGH:<GM-"BGHBRkLKJ./.3:<LB<@[email protected]"F=<-N02B<-*:<Ll"-"=<La02B:<,.-m 0=?02Ca-:<-"=

RkJ.=<:<,.-=6GM/.F"=<-N02B<-"B?XNGM/T3.GMFN02:<GH/.UB2:?02W.GH@MGH:AVRkL;=*GM/N:<-"/.B<--"@M-"F":<=<LK/B2m.GM=?0@8:<L;=<LKGH3.B?)

xB<-GMB0@MB<LaC03.-L;R*(*ITS ` E?Sj:<LaGM/NdK-"B<:<GHUT02:<-:<,.-B<:?02W.GH@MGH:QVam.=<L;mT-"=<:<GM-"B*L;R+GH/N:<-/.B2- ` EKSj-yNJ.GHg@HGMW.=<GQ0GH/q:<,.-502W.B<-"/.F"-LKRa02/NVs02mTm.@MGH-3:<LK=<LKGH3 02@6CD02UK/.-":<GHFuT-@H3 ) e @H@*L;R:<,.-GM/N:<-"/.B<- ` E?S*Ba_-

GH/;d;-B2:<GMUT0:<-"30=<-*RkLKJ./.3a:<LW8-*B<:?0W.@M-N) e B0:AVNm.GHF;[email protected]"v80Cm.@H-NX?:<,.-6=<-B2J.@M:<BL;RT:<,.-*B2:?02W.GH@MGH:AV02/ 0@MVNB<GHB

02=<-B<JTCCD02=<GHl-"3GH/rITGHUT) *RkL;=:<,.-GM/N:<-"/.B<- ` E?S-"yNJ.GH@MGHW.=<GMJ.CB<,TL?_*/GH/rITGMU8)T\6GH/rE?-"F;);P<P")TITGHUKJ.=<-"B

[K\

Page 15: Equilibrium and Stability Properties of Self-Organized Electron ...

Kn<0;p02/.3 KnkW p+B<,.LK_XK=<-"B<m8-F":<GHdK-"@MVNX;:Q,T-6/.L;=<C0@MGHl-"3=<-N02@TRk=<-"yNJ.-/.F"VD4- n "! p02/.3/.LK=<CD02@HGHl-"3GHC0UKGH/ 02=<VDRk=<-"y;JT-/.F"VDPQC n "! p0B0RkJ./.F":<GHLK/rL;R:<,.-am8LK@HLKGH3 [email protected]/;JTCWT-"= RkL;=:<,.-a-"vNg:<=?02L;=<3.GM/ 0=<VD-GHUK-"/.CaLN3.-amT-"=<:<J.=<W 02:<GHLK/.B?)aS*,.-a=<-N02@Rk=<-yNJ.-"/.F"VDGM/.F"=<-N02B<-"B02B:<,.-aCaLN3.-a/NJ.CWT-"=

GH/.F=<-N02B2-BK)+S*,.-*GMCD02UKGH/ 0=QVm 0=<:<BLKRT:<,.--"GHUK-"/.Rk=<-yNJ.-"/.F"GM-"B602=<-*-"GM:<,.-"=l-"=<LL;=/.-U802:<GHdK-N)+S*,.-"=<-"RHL;=<-;X

02@H@+LKR:<,.--"GHUK-"/.CLN3.-"B02=<-B2:?02W.@H-;)PA:6GMB6_LK=<:<,N_*,.GH@H-/.LK:<GH/.Ua:<, 02:6:<,.-"=<-02=<-CL;=<--"GMU;-/.CaLN3.-B6GH/:<,.-GH/;:<-"/.B<-F;0B<-:<, 02/GH/:<,.-:<-"/NJ.LKJ.B*FN02B<-N) S*,.-CaLN3.-B*_*GH:<,,.GHUK,.-"=*RH=<-"yNJ.-"/.FGH-"BF"LK=<=<-"B<m8LK/.3:<L

,.GHUK,.-"=<giLK=<3.-"=*=?03.GA0@+LKB<F"GM@H@Q02:<GML;/.B?)

[ b

Page 16: Equilibrium and Stability Properties of Self-Organized Electron ...

%a% cr-, 0<d;-m.=<-B2-/N:<-"3j0aFL;@M3.gAh.J.GM35CLN3.-"@RHL;=0aB<-"@MRkgiL;=<UT02/.GHl"-35-@H-"F:<=<L;/1B<m.GH=?02@:<LK=<L;GM3sn ` E?Sp

GH/5m 02=<:<GQ02@0:<CL;B<m.,.-"=<-NX*0B<B<J.CaGM/.U0Kp6:<,.--@H-"F:<=<L;/.BJ./.3.-"=<UKLr-/.-"=<UK-":<GHFB<mTGM=?02@*CaLK:<GML;/s02@HLK/.U0

,.LK@H@HL?_j:<LK=<J.B*_6GM:<,10/GML;/W 02F9;U;=<LKJ./.3 XNW p :<,.--@H-"F:<=<L;/C-N02/Rk=<-"-m 02:<,GHB*@ML;/.UF"LKCam 02=<-"3_*GH:<,

:<,.- ` E?S B<GHl-NX+F;p:<,T--"@H-F":<=<LK/1:<-CamT-"=?0:QJT=<-aGMB@ML?_X+3 p:<,.-aGHLK/1W 02F29NU;=<LKJ./.31GMB02B<B<JTC-"3r:<LWT-u.vN-"3 X 02/.3D-Np:<,.-CaGM/.L;=6=?03.GMJ.BLKR:<,.- ` E?S'GMBB<CD02@H@ F"LKCam 02=<-"3_*GH:<,:<,T-C0?wkL;=6=?023TGMJ.B?)xB<GH/.U

:<,.GHBCaL;3.-"@QXK:<,.--"yNJ.GM@HGMW.=<GHJ.Cz02/.3B2:?02W.GH@MGH:AVam.=<L;mT-"=<:<GM-"BL;R8:<,T-6-"@M-"F":<=<LK/h.L?_jGM/:<,.-B<-"@HRkgiLK=<U802/.GHl-"3

` E?Ss_-=<-0/ 02@HV;l"-"3 )

PQ/qm 0=<:<GMF"J.@Q02=?X:<,.--"y;J.GH@MGHW.=<GHJ.C&mTL;@ML;GM3 0@6-@H-"F:<=<L;/qh.L?_ d;-@HLNFGH:AV'02/.3-yNJ.GH@MGHW.=<GMJTC&:<LK=<LKGH3 02@B<-"@MRkgAC0UK/.-":<GMF6u.-@H3a_-=<-63.-:<-"=<CaGM/.-"3aRkLK=0B<CD02@H@Mg02B<m8-F":=?02:<GML,.LK@H@ML?_ ` E?S) ` v+02Cam.@M-"BLKR8GH/;:<-"/.B<-

02/.3r:<-/NJ.L;J.B ` E?So-"y;J.GH@HGMW.=<GQ0_-"=<-m.=<-B2-/N:<-"3 )axB<-_02BCD023.-L;R:<,.-F"LK/N:<GM/NJ.GH:AV-"yNJ 02:<GHLK/ XT:<,.-

RkLK=<F"--"yNJ 02:<GHLK/ X0/.3 O 0v;_-"@M@ Ba-yNJ 0:<GMLK/TBa:<LrLKW.:?0GM/'0F"LKCam.@H-:<-B<-":aLKR@MGH/.-N02=<GHl-"3q-"y;J 0:<GML;/.B

UKL?d;-=</TGM/.U6-"v;:<=?0LK=<3.GH/ 02=<VCaLN3.-mT-"=<:<J.=<W 02:<GHLK/.BK) e /-"GHUK-"/;[email protected] 02:<GHLK/RkLK= -vN:<=?02L;=<3.GM/ 0=<VCLN3.-

m.J.=<:<J.=<W 0:<GML;/.B_0B3.-"=<GMdK-"3 X02/T3qB2LK@Md;-"3q/NJ.Ca-=<GHFN02@H@MVq_*GM:<,o0F"LKCam.J.:<-"=aF"LN3.-N)PQ:a_0BaRkL;J./.3:<, 02:0aF"@Q02B<BL;R*B<-"@HRkgiLK=<U802/.GHl-"3 ` E?S-"y;JTGM@HGMW.=<GQ0a-"v;GHB<:<B_*GH:<,.LKJ.:0/1-"vN:<-=</02@[email protected].@HGM-"35:<LK=<LKGH3 02@

CD02UK/T-:<GHFu.-@H3 )PA:*_0B*B<,TL?_*/:<, 0:*GM/:<,.-02W.B2-/.F"-LKR602/NVr02m.m.@HGH-3:<LK=<LKGH3 02@+CD02U;/.-:<GHFu.-"@M3 XN:<,.-

` E?S5-"yNJ.GM@HGMW.=<GQ002=<-*B<:?02W.@H-0: ,.GHUK,-"@M-"F":Q=<L;/3.-/.B2GM:<GH-B6nkGQ) -N)HXN0: ,TGMU;,:<L;=<LKGH3 02@NB<-"@HRkgiC0UK/.-":<GHFu.-"@M3.B?p<X

02@H:<,.LKJ.U;,:<,.-V02=<-*J./.B<:?02W.@H-0: @HL?_5-"@M-"F:<=<L;/3.-/.B2GM:<GH-B6nkGQ) -N)HX.02:@ML?_5:<LK=<L;GM3 0@NB<-"@HRHgAC0UK/.-":<GMF*u.-"@M3.B?p<)

[K

Page 17: Equilibrium and Stability Properties of Self-Organized Electron ...

6*$%a2*%

o sx/.3.-"=:<,T-6-"vN:Q=?0LK=<3.GH/ 02=<VNgiCaLN3.-6m8-"=Q:<JT=<W 02:<GHLK/.BU;GMdK-"/GH/ ` yNB?)Tn<[;fKpkg2n<Z;YKp<X;:<,.-6F"LK/TB<-=<d.0:<GML;/LKR

-"@M-"F":<=<LK/F, 02=<UK-GHB*CD02GH/N:?02GH/.-3X;_6,.GMF,GMB63.-"B<F"=<GMW8-3W;V:<,.-F"LK/N:<GH/;JTGM:QV-yNJ 0:<GMLK/

99:;

n# ; p .=<?> n# ; p)* n# ; p sY n [Kp

NGH/.-N02=<GHlGH/.U ` y+)8n e [Kp<XN_-LKW.:?0GM/

8 . [ n * p . * + . * + sY n ZKp

IK=<LKC ` y+)+n e Z;p<X;:<,T-3T-/.B2GM:QVmT-"=<:<J.=<W 02:<GHLK/ GMB6UKGHdK-"/WNV

[ 8 8

n * p .

* + n \Kp

_*,.-"=<-:<,.--"y;J.GH@HGMW.=<GHJ.Ct-@H-"F:<=<L;/102/.U;J.@A0=*h.L?_sdK-"@MLNF"GM:QV GMB63.-"u./.-3W;V

2n p *,+ n p n b p

_*,.GHF2,GMB06RkJ./.F":<GHLK/L;R )S*,.-@HGH/.-;0=<GMl"-"3RkLK=<F"--yNJ 0:<GML;/GMB6UKGHdK-"/WNV

99#; )*0. n * + )- + >< p )* . n )* >< p * + )- + 8 -

n ) .0* + )- + ) . )* 1 )- 1"p n Kp

` yNJ 02:<GHLK/5n e ;p F;0/W8--vNm.=<-"B<B<-"3GH/:<,.-F"LKCamTL;/.-"/;:*RkL;=<C

8 n 8 2p * 8 Z * + . * + 8 -n .0* + 1"p6 n Kp

8 n 8 2p * + . [ n

2p 8 * 8 -

+ n fKp

_*,.-"=<- *n p -

1 n p . n Kp

GHB:<,.--@H-"F:<=<L;/F"V;F"@HLK:<=<LK/Rk=<-"yNJ.-"/.FVaGH/:<,.--"y;J.GH@MGHW.=<GHJ.C :<L;=<LKGH3 02@8CD02UK/.-":<GHF6uT-@H3 1 n p . )- 1 )[

Page 18: Equilibrium and Stability Properties of Self-Organized Electron ...

S*,.-m8-=<:<J.=<W8-35-@H-F":<=<GHFD0/.31CD02UK/.-":<GHFau.-"@H3.B ) n ; p02/.3 ) n ; p*B?02:<GHB<RkVr@MGH/.-N02=<GHl-"3O 02vN_-@H@ B6-yNJ 02:<GHLK/.B

< > ) n# ; p 8 - 2n# ; p6 n ]Kp

< ) n# ; p . 99#; ) n# ; p sY( n [;YKp

< > ) n# ; psY, n [;[Kp

< ) n# ; p 8 [ 99:; ) n ; p 8 # - n p )* n ; p .0* +Nn p )- + 2n# ; p n [;ZKp

LK:<-:<, 02:6RHL;= ) n# ; p 1 n p - 2 + )- 1 X ` y8)+n e [K[KpGMB02J.:<L;C0:<GMFN02@H@MVB?0:<GMB2u.-3X;GQ) -N)HX

<?> ) n# ; p [ 9

9 n p 1?n p - 2 + sY, n [;\Kp

_*,.-"=<- GMB:<,.-CD0?wkL;= ` E?S'=?03.GMJ.BKX02/.3 GHB:<,.-3.GHB<m.@Q02F"-"C-"/N:a02@HLK/.U:<,.-:<LK=<LKGH3 )PA:B<,.LKJ.@H3WT-a=<-"CGH/.3.-"3r:<, 02::<,.-aJ./.GH:dK-"F:<L;= )- 1 GMBFL;/.B<GH3.-=<-"302B0F"LK/.B<:?0/N:J./.GM:d;-F":<LK=GH/r:<,.-am.=<-B2-/N:

02/ 0@MVNB<GHB?) ` y;J 0:<GML;/.Bn e ]Kp<X+n e [KYKp602/.31n e [KZ;p FN02/W8--vNm.=<-"B<B<-"3GH/:<,.-F"LKCamTL;/.-"/;:6RkLK=<C

[ n p . + 8 -

n [ b p

[ n +Np 8 1 n [;Kp

1 . 8 # - * n [ Kp

8 1 .

+ 8 - n *(+/.0*,+ p n [;fKp

c,.GH@M-:<,.-"=<-02=<-RkLKJ.=-"yNJ 02:<GHLK/.Bn e [ b pkg2n e [KfKpRkL;=:<,.=<-"-u.-"@H3Dd.02=<GQ02W.@H-"B X + 02/.3 1 XT_-_LK=<9_*GH:<, ` y;B?)+n e [K;p*02/.35n e [ Kp :<LaLKW.:?0GM/

8 [[ 8

.

-

* n [;Kp

02/.3

1 [[ 8

.

-

* n [;]Kp

[Kf

Page 19: Equilibrium and Stability Properties of Self-Organized Electron ...

_*,.-"=<- 6n p GHB*3.-"u./.-"3GH/ ` y+)+n<Z b p<) L;:Q-:<, 0:*_*,.-"/ * * [KXN_-, 0<dK-0m.m.=<L?vNGMCD02:<-"@HV

8 # + 8 1 Y, n Z;YKp

F"LK=<=<-"B<mTL;/.3.GH/.U:<L *m8-=<:<J.=<W 0:<GML;/.B?)S LaB<J.CaCD02=<GHl-W.=<GH-"h.V;XN-"yNJ 02:<GHLK/.Bn<Z b pQX+n e \Kp<X+n e Kp<X+n e fKp<X+n e [;Kp*0/.31n e [K];p =<-"@A0:<-:<,.-mT-"=<g

:<J.=<WT-"3ayNJ 0/;:<GH:<GM-"B X * X *,1 X X + X802/.3 1 ) S*,.-"B<-6-"y;J 0:<GML;/.B:<L;UK-":<,.-=_*GM:<,D7TL;GMB<B2LK/ B

-"y;J02:<GML;/5n e [ b p_*GH@M@+W8-J.B2-3GH/ e m.m8-"/.3.GMv11:<La3.-"=<GHdK-0/-GHUK-"/[email protected]"yNJ 02:<GHLK/GH/;d;LK@Hd;GH/.UaL;/.@MV

6n pjn # p +Nn p<)

[K

Page 20: Equilibrium and Stability Properties of Self-Organized Electron ...

*66 o # 6 S*,.--"vN:<=?02LK=<3.GH/ 02=<VNgiCaLN3.--GHUK-"/[email protected]"yNJ 02:<GHLK/DGHB3.-"=<GMd;-3rGM/D:<,.=<-"-C0?wkL;=B<:<-"m.B?) e Ba0u.=<B<:

B<:<-"m XN_--@HGMCaGH/ 02:<-:<,.-3.-"/.B<GH:AVm8-=<:<J.=<W 0:<GML;/ GH/ ` y+)+n e [ b p WNVB<J.W.B2:QGH:<J.:<GH/.U ` y;BK)8n e \Kp*0/.3

n e [KKpGM/N:<L7TL;GMB<B2LK/ B*-"yNJ 02:<GHLK/1n e [ b p<) S*,.GHB*=<-N023.GH@MVU;GMd;-B

[

[ 8

8

- 8

8 [ n * p 8 [

[ 8 n * p . 8 * + n [Kp

PQ:GHB-"dNGM3.-"/N:GH/ ` y+)*n<[;p6:<, 0::<,.-u.-"@H35mT-"=<:<J.=<W 0:<GML;/ #sGHB3.=<GHdK-"/qB<-@HRkgiF"LK/.B2GMB<:<-"/N:<@MV1WNV1:<,.--"@M-"F":<=<LK/3.-"/.B<GH:QVmT-"=<:<J.=<W 02:<GHLK/_*,.GHF2,GHB6/.L?_j-"vNm.=<-"B<B<-"3GH/:<-"=<CB6LKR:<,.-dK-"@MLNF"GM:QVm8-=<:<J.=<W 0:<GML;/.B

* 0/.3 *,+ )e B06B<-"FL;/.3B<:<-"m X;_--vNm.=<-"B<B*:Q,T-6d;-"@MLNFGH:QVamT-"=<:<J.=<W 02:<GHLK/.B * 02/.3 * +GM/:Q-"=<CaBL;R+:<,.-u.-@H3

mT-"=<:<J.=<W 0:<GML;/ DJ.B2GM/.U:<,T-*@HGM/.-N02=<GHl"-3aRkL;=<F-6-"y;J 0:<GML;/.Bn e ;p02/.3Dn e fKp02/T3a@MGH/.-;0=<GMl"-"3 O 0vN_-"@H@ B-"y;J02:<GML;/.Bn<Z b p<X8n e [K;p*02/.31n e [K];p<) PA/.3.-"-"3 X;B2J.W.B<:<GH:<J.:<GM/.U ` yNB?)8n e [K;p*02/.31n e [K];p+GH/N:<L ` y+)+n e Kp<X

B<L;C-B<:<=?0GMU;,;:<RkL;=<_02=<310@MU;-W.=?0VNGM-"@M3TB

n 8 2p [ .

[ 8 * . n 8 Z p * + -

[ 8 *(+ # [ 8

# n ZKp

02/.3B<J.W.B2:QGH:<J.:<GH/.U ` y8)+n<Z b p GH/;:<L ` y+)+n e f;p VNGM-"@M3.BKX

[ n

2p 8 * . n 8 2p * + 8 -

n \Kp

E?LK@HdNGM/.U ` yNB?)+n<*ZKp*0/.31n<*\Kp RkLK= * 02/.3 * +NXN_-u./T3:<, 02:

* -

n 8 Z p . n 8 p [ 8 * +# [ 8

n b p

*,+ -

[ 8 * +# [ 8

[ n

p 8

8 n 8 p [ .

[ 8

n Kp

_*,.-"=<-:<,.-yNJ 02/N:<GM:QV n p GHB*3T-u./.-"3GH/ ` y8)+n<ZKf;p<)

[K]

Page 21: Equilibrium and Stability Properties of Self-Organized Electron ...

e B06:<,.GM=<3B<:<-"m X;_-B2J.W.B<:<GH:<J.:<- ` y;BK)Tn< b p*02/.3rn<*Kp+GH/N:<L ` y+)8n<[;p+:<LLKW.:?02GH/:<,.--"GMU;-"/;[email protected]

-"y;J02:<GML;/

[

[ 8

# 8

[ 8 [ .

[ 8 #

[ 8

[

n 8 Z p # . n 8 p [ 8 *(+ #

[ 8 # n Kp

. [

[ 8 n 8 Z p . n 8 p [ 8 * +#

[ 8

.

[ 8

[ 8 *(+ # [ 8

[ n

2p 8

_*,.-"=<- [, Z, >>> X802/.3 n p - n p6 ! GMB*:<,.-ank@MLNF;0@Ap+-"@H-F":<=<LK/m.@A0B<C06Rk=<-yNJ.-"/.F"V;)ITGH/ 02@H@MVNX0Rk:<-=0@M-"/.UK:<,NV50@MU;-W.=?0GMFa-vN-"=<FGHB<-NXT_-FN02/r-"v;mT=<-B<B:<,.--"GHUK-"/;dT02@MJT--yNJ 0:<GML;/qn<Kp

GH/:<,.-RkLK@H@ML?_6GM/.UaRkLK=<C

[

[ 8 M[ . n p 8

H[ . +Nn p sY( n fKp

_*,.-"=<-*:<,.-6B<J.B<F"-"m.:<GMW.GH@MGH:QVRkJ./.F":<GML;/.B n p0/.3 + n p02=<-*3.-"u./.-"3aGM/ ` yNB?).n<Z;Kp02/.3Dn<Z Kp<X=<-"B<mT-"F":<GMd;-@HVN)

` yNJ 0:<GMLK/5n<*fKpCJ.B<:6WT-B<LK@HdK-"3B2J.W+wk-F":*:<LD0m.m.=<LKm.=<GQ02:<-WTLKJT/.3 02=<VF"LK/T3.GM:<GHLK/.B?)I?L;=0/ ` E?S

GH/LKm8-/B<m 0F-nk-N) U8)MXNGH/102GH=?p<XN:<,.-WTLKJT/.3 02=<VF"LK/.3.GH:<GML;/.B0=<-

#*n7sYKpsY 02/.3 #*n7 p ! n Kp

( L;/.B<GMB2:<-/N:_*GH:<,r:<,.-aW8LKJ./.3 0=<VDFL;/.3.GM:<GHLK/.BGM/ ` y+)n<*Kp<X+:Q,T-W8LKJ./.302=<VDFL;/.3.GH:<GML;/.B02: "!02/.3 _6GM@[email protected]"=<GHdK-"3GM/ e m.mT-"/.3.GHv ^ 02Rk:<-"=0*W.=<GH-R+3.GHB<F"J.B<B<GHLK/L;R+:<,.-6B2LK@MJT:<GMLK/TB:<L ` y+)8nQ*fKp+GH/

:<,.-d.02F"J.J.Ct=<-"UKGHLK/.B6GM/ e m.m8-/T3.GMvr(*)

ZKY

Page 22: Equilibrium and Stability Properties of Self-Organized Electron ...

% 2& 6a% #cr-B<LK@HdK-:<,.--GHUK-"/[email protected]"yNJ 02:<GHLK/'n<*fKpGM/q:<,T-d.0FJ.J.C&=<-"UKGHLK/.Ba_*GH:<,0 % "! 02/.3q_*GH:<,

XN_6,.-=<- n p 2n p * +Nn p * n p 'YK)IKLK= % "! L;= X ` y+) n<f;pFN02/-"v;mT=Q-"B<B2-3502B

[

[ 8

8

sY n [Kp

LK= # . [

[ . [ 8

# .

8

# sY n ZKp

c,.GH@M-/.-GH:<,.-"= ` y+)8n<(6[Kp+/TLK= ` y8)+n<(*ZKp B<-"-CaB*:<LWT-02/ 02@HVN:<GMFN02@H@MV:<=?02F":?02W.@H-;XN_-FN02/B<LK@HdK-RkL;= # 6WNV=<-:<J.=</.GH/.Ua:<L ` yNB?)+n e [ b pkg2n e [;fKp<)6E?-":<:<GH/.U 2n p n p * +Nn p * n p sYKX

_--vNm.=<-"B<B ` yNB?)+n e [ b pkg2n e [Kf;p*02B

[ n p . + jY( n \Kp

[ n + p 8 1 n b p

1 . sY( n Kp

8 1.

+ jY n Kp

cr-, 0<d;-Rk=<L;C ` y+)8n<(6\Kp<X

+ 8 [ n p6 n fKp

02/.3Rk=<LKC ` y+)+n<(*Kp<X

1 8

n Kp

E?J.W.B2:QGH:<J.:<GH/.U ` y;BK)8n<(6fKp*0/.31n<(*;p GM/N:<L ` y+)8n<( b p [email protected] -"B<B<-"@ B*-"yNJ 02:<GHLK/

[

n p . 8

n p sY n ]Kp

RkLK=6:Q,T-d.02=<GA0W.@M- )-"/.-"=?02@+B<L;@MJ.:<GHLK/.B6:<L ` y8)+n<(*];p*02=<-

n p @ n # p . n # p" n [;YKp

ZK[

Page 23: Equilibrium and Stability Properties of Self-Organized Electron ...

_*,.-"=<- @ n p60/.3 n p602=<-:<,.-u.=<B<:<g0/.3B<-"FL;/.3.gi9NGM/T35+-"B<B<-"@ RHJT/.F:<GHLK/.BLKRLK=<3.-"= X+0/.3 0/.3

0=<-aF"LK/.B2:?02/N:<B:<LW8-a3.-":<-"=<CGH/.-"3rWNVD:<,.-WTL;J./.3 02=<VrFL;/.3.GM:<GHLK/.BU;GMd;-/1GH/ ` y+)n<*Kp*:<LKU;-:<,.-"=

_*GH:<,qB<L;@MJ.:<GHLK/.B:<L ` y8)6n<*fKp_*GH:<,.GM/:<,.- ` E?St-"@H-F":<=<LK/@A0<V;-=Dn"! % % p<)'E?J.W.B<:<GH:<J.:<GH/.U

` yNB?)8n<(*fKp02/.3rn<(6[KYKp+GH/;:<L ` y+)Tn<Z b p<XK_-LKW.:?0GM/U;-/.-"=?02@8B<LK@HJ.:<GHLK/.B:<L:<,.--"GMU;-/[email protected] 02:<GHLK/GH/

:<,.-d.02F"J.J.Ct=<-"UKGHLK/.B6LKJ.:<B<GH3.-:<,.- ` E?Ss-@H-"F:<=<L;/@Q0<VK-"=*_*GH:<, '%&"! 02/.3_*GM:<, X

6n p 8 @ " n # p . " n p n [;[Kp

_*,.-"=<- @#" n p @ n p6 02/T3 " n p n p6 )cr-D/.L?_ u./.3jB<mT-"F"GA0@B<LK@HJ.:<GML;/.BRHL;=r0/ ` E?S GH/LKm8-"/B<m 0F-DWNVq:?09;GH/.U5GM/N:<Ls02F"F"LKJ./N::<,.-

WTL;J./.3 0=QVF"LK/.3.GH:<GML;/.BUKGHdK-"/aGH/ ` y+).n<;p<)+PQ/a:<,.-6GM/./.-"=d.02F"J.J.C =<-U;GML;/a_*GM:<,DY '%&"! X sYKX

WT-"FN02J.B2-0@M@8LKR+:<,.-u.-"@M3mT-"=<:<J.=<W 02:<GHLK/.BCJ.B<:W8-6u./TGM:<-a02:7sYK) S*,T-=<-"RkLK=<-NX;RkLK=Y '%&"! X;:<,.--"GMU;-/.RkJ./.F":<GHLK/ #*n p FN0/WT--"vNm.=<-B2B<-"3102B

#*n p 8 @ " n # p6 n [;ZKp

02/.3:<,.-F"LK=<=<-"B<mTL;/.3.GM/.Uam8-=<:<J.=<[email protected]=<-UKGHdK-"/WNV

1 n p 8

@ n # p6

n p @ n p6 n [;\Kp

+ n p

@ " n # p~-=<-NX

GHB0F"LK/.B<:?0/N:*:<LW8-3.-":<-=<CaGM/.-"3W;V:<,.-W8LKJ./.3 0=<VFL;/.3.GH:<GMLK/TB02: 7 "! )

PQ/:<,.-L;J.:<-"=*dT02F"J.J.Ct=<-"UKGHLK/_6GM:<, X X.WT-"FN02J.B2-:<,.-mT-"=<:<J.=<WT-"3u.-"@M3TB6RkLK=<C 02/

LKJ.:<gAUKLKGH/.Ua_0<dK-N) S*,.-"=<-"RkLK=<-NX;RkL;= XN:<,.--"GMU;-"/.RkJ./.F:<GHLK/ *n pF;0/W8--vNm.=<-"B<B<-"310B

6n p 8 $ " n # p6 n [ b p

_*,.-"=<- $ n p GHB*:<,.-u.=<B2:<gi9NGM/.35~60/.9K-"@8RkJ./TF:<GHLK/LKR LK=<3T-= 3.-"u./.-"3WNV

$ n p @ n p . n p n [;Kp

ZKZ

Page 24: Equilibrium and Stability Properties of Self-Organized Electron ...

S*,.-F"LK=<=<-"B<m8LK/.3.GH/.UamT-"=<:<J.=<WT-"3u.-"@M3TB02=<-U;GMd;-/W;V

1?n p 8

$ n # p"

n p

$ n # p6 n [ Kp

+Nn p $ " n # p

~-=<-NX GHB0F"LK/.B<:?0/N:*:<LW8-3.-":<-=<CaGM/.-"3W;V:<,.-W8LKJ./.3 0=<VFL;/.3.GH:<GMLK/TB02: 7 )S*,.-=<CaB6=?023.GQ02@7TL?V;/N:<GH/.Uh.JTvGMB

n p [Z + n p 1 n p?-"v;m n<ZPAC ; p 8 Z

* * * *

$ " n p $ n # p?-vNm n<ZPAC ; p6n [;fKp

RkLK= )PAR:<,.- ` E?SsGMB6B<:?0W.@M-_*GH:<,PQC YKX.:<,.-/:<,.-7.L?VN/N:<GM/TUah.J.vGHB6B<CD02@H@+J./.3.-"=6B<CD02@[email protected].=<:<J.=<W 0:<GML;/.B?)/:<,.-LK:<,.-"=*,02/.3 XNGHR :<,.- ` E?SsGMB6J./.B<:?0W.@M-_6GM:<,PAC YKXN:<,.-"/:<,.-7.LKV;/N:<GH/.U

h.J.vGH/.F"=<-;0B<-"B -"vNmTLK/T-/N:<GA0@M@HV02B60RkJ./.F":<GML;/LKRT:<GMCa-02Bm.J.=<:<J.=<W 0:QGHLK/TB U;=QLK_X"=<-"3.J.F"GH/.U6:<,.-@HGHRH-":<GHC-

LKR:<,.- ` E?S)

ZK\

Page 25: Equilibrium and Stability Properties of Self-Organized Electron ...

*66 oa%S LB<LK@HdK-:<,.--GHUK-"/[email protected]"yNJ 02:<GHLK/qn<*fKp/NJ.Ca-=<GHF;0@M@HVGM/D:<,.--"@M-"F:<=<L;/D@A0<V;-"=6Rk=<L;C = "! :<L

X _-CJ.B<:B<-"::<,.-W8LKJ./.3 0=<VrFL;/.3.GM:<GHLK/.B02::<,.-GH/./.-"=02/.35LKJ.:<-"=B<J.=<R<0F-"BL;R02/ ` E?S)IKLK=

m.=<-"B<-"/;:m.J.=<mTL;B<-"B?X"_-GH/N:Q=<LN3.J.F"-:<,.-023TCGH:<:?02/.F"-3.-"u./.-3GM/ ` y+);n<Z;]Kp<X?_*,.GHF2,GHB :<,.-=?0:<GMLWT-":Q_--"/

:<,.-=?023TGA02@*7TL?VN/N:<GM/.Uh.L?_'mT-"=6J./.GH:6@H-/.U;:<,GH/:<,.- gA3.GM=<-"F":<GMLK/q02/.3D:<,.-B<yNJ 02=<-L;R:<,.-nk-T-"F:<GHdK-NpdKL;@M:?0UK-02B2B<L;F"GQ02:<-"3a_*GH:<, + )E?GH/.F"-*W8LK:<, + n p0/.3a:<,.-6=?023.GQ02@7TL?V;/N:<GH/.Uh.JTv0=<-6FL;/;:<GH/NJ.LKJ.B?XGH:*RkLK@H@ML?_*B6:<, 02:6:<,.-023.CaGM:<:?0/.F- GHB 60: 7 "! 02/.3 7 )

IKLK=0/ ` E?SGM/L;mT-"/B<m 0F-ank-;) UT)HX;GM/r02GH=?p<X;:<,.-6W8LKJ./.3 0=<VF"LK/.3TGM:<GHLK/.BFN02/W8-63.-"=<GHdK-"3_*GM:<,:<,.-

02GH3L;R+:<,.-B2mT-"FGQ02@8B2LK@MJT:<GMLK/TBL;W.:?02GH/.-3GM/ e mTmT-"/.3.GMvr(0/.3:<,.-F"LK/N:<GH/;JTGM:QVaLKR :<,.-a023.CaGM:<:?0/.F"-a02::<,.- ` E?SsB<J.=<R<02F"-"B?)*E?J.W.B2:<GM:<J.:<GH/.U ` y8)+n<(*[;\Kp GH/N:<L ` y8)+n<ZK;p<X+02/.3B2-:<:<GH/.U 7 "! 8 YKXN_-LKWT:?02GM/

?n7 "! 8 YKp Z! "! @ n "! p@ " n "! p n [Kp

E?GHCGH@A0=<@MVNXNB<J.W.B<:<GH:<J.:<GH/.U ` y+)+n<(*[ Kp GH/N:QL ` y+)+n<ZK;p<X80/.3B<-":<:<GH/.U 7 . YKXN_-LKW.:?0GM/

n7 . Y;p Z # $ n p$ " n p n ZKp

` dNGM3T-/N:<@HV;X;LK/.-LKR+:<,.-a023.d.0/;:?0UK-"BLKR+GH/N:Q=<LN3.J.F"GH/.U:<,.-a023.CaGM:<:?0/.F- GHB:<, 0::<,.-F"LN-aF"GM-"/;:<B

02/.3 3.L/.L;:02m.m8-N02=*-"vNm.@MGHFGH:<@HVGM/ ` y;BK)8n ^ [Kp*0/.31n ^ Z;p<)

PQ/.B<GH3.-:<,.- ` E?S -@H-"F:<=<L;/@A0<V;-=?X0/o02/ 0@MVN:<GHF;0@6-"vNm.=<-B2B<GML;/RkLK=:<,.-q023.CaGH:Q:?0/.F"- F;0/WT-

LKW.:?0GM/.-"3W;VB<J.W.B<:<GH:<J.:<GM/.U ` yNB?)+n<Z b p<X+n e ZKYKp602/.35n< b p GH/;:<L ` y+)+n<ZK;p<) S6,.GMB6UKGHdK-"BnkRHL;= "! p

?n p Z! [

[ 8

8

[ n 8 Z p .

8

n 8 p [ 8

[ 8 [#

# n \Kp

E?GH/.F- n p GMB6F"LK/N:<GM/NJ.LKJTB02: 7 6! 02/.3 XN_-LKW.:?0GM/:Q,T-RHL;@M@HL?_*GH/.UW8LKJ./.302=<VF"LK/.3.GH:<GML;/.B n7 "! p Z # "!

@ n "! p@ " n "! p n b p

02/.3?n7 p Z!

$ n p$ " n p n Kp

Z b

Page 26: Equilibrium and Stability Properties of Self-Organized Electron ...

_*,.-"=<- n pGMB3.-"u./.-"3rGM/ ` y+)n<ZK]KpRkLK= "! % &% )S*,.GHBFL;/.F"@MJ.3.-"B:<,.-a3.-=<GHd.02:<GHLK/rL;R:<,.-WTL;J./.3 0=QVF"L;/.3.GM:<GHLK/.B6GM/ ` yNB?)+n<ZKZKp602/.31n<Z;\Kp<)

ZK

Page 27: Equilibrium and Stability Properties of Self-Organized Electron ...

a% # 'S*,.GHB_L;=<95_02BB<J.m.mTL;=<:<-3GM/m 02=<:aWNVs*02@H@MGHB<:<GMF O GHB<B<GH@M- ^ -"Rk-"/.B<- =<UT0/.GMlN02:<GHLK/ X( LK/N:<=?02F":

LT) ^ E?c e Y;[2g2];f2g O g2Y;K\Kf;) S*,.- e JT:Q,TLK=<B_6GMB<,s:<L5:<, 0/.9 ^ =?) cGM@H@MGQ02C J.B2BRkLK=mT=<L?d;GH3.GH/.U

,.-"@Mm.RkJ.@+F"LKCaCa-/N:<B0WTL;J.:*:<,.GHB*m 0mT-"=?)

Z

Page 28: Equilibrium and Stability Properties of Self-Organized Electron ...

*6%a

[K)78) O ). -@H@Q02/ X nkPQCm8-=<GQ02@+( LK@H@M-"UK-78=<-"B<B?X ! ZKY;YKY;p+0/.3=<-"Rk-=<-"/.F"-B :<,.-"=<-"GM/ )

ZK)E.)+NJ./T3.y;J.GHB<:?X e =<9;GHdRkLK=ITVNB<GH9 #" X8\ K[an<[K];KYKp<)

\K)~)878)+IKJ.=<:<, X O ) e )+N-"dNGM/.-NX+02/.3546);co)Nc0/.GM-"9+X+4-d+)*E?F"GA)PA/.B2:<=<J.C) "$ X8] b ]an<[K];KfKp<)

b )+)Tco).crLK@H:<wk-"="X78=<LNFN) 02:?) e FN023 )E?F"GA)n<x6E e p&%%8X b ]KYn<[K];KKp 7+=<L;FN) 02:?) e FN03 )E?F"GA)n<x6E e p%%8X+;\K\an<[K]K;Kp

K) )+)NS*0<VN@HLK=?X+78,NV;BK)*4-"d8)*.-":<:?)''8X+[K[K\;]n<[;]Kf b p<)

K)co)+~6)+ LKB<:<GHF29+X+E?F"GA) e Ca-=?)()*8X+;fn<[;]KKf;p<)

fK)~) e @MRkd+-"/ X++)+NGH/.3.W8-=<U8X80/.3178) O GM:<@HGH3 X ) J.F"@A) ` /.-=<U;Vrn<702=<:(*p&(8X8[;[ an<[K];K]Kp<)

K) ^ )+46)Ncr-@H@HB?X+78,;VNB?)[email protected]*8X+;Z an<[K] b p<)

]K)46)+4602CD02/ XN-":02@Q)MX+78,NVNB?)*4-"d+)*N-":<:?)*'8X+\K[;YK[an<[K]K];\Kp<)

[;YK) )8~)8~602CaCa-=?X+(*);co)8~60=<:<C0/ X )8 ) ` 3.3.@M-"CD02/ X80/.3r~)8E.) O F;.-N02/ X878,NVNB?)*4-"d8)*.-":Q:?), (8X+ZK b \an<[K]K;Kp<)

[;[K) )+~6) ^ -"UK/02/ XN-":02@Q)MX+78,NVNB?)*[email protected] #- X8Z;]K\;n<[;]K]K\;p<)

[;ZK)E?-"-;XNRkLK=*-"v+02Cam.@H-;X O ) e )+xCD02/ X./0 1# n e FN023T-CaGMF78=<-"B<BKX -_32L;=<98X[K];KfKp<)

[;\K) ^ ).(*)TE?-_0=<3 X.(*)T( ,.-"/ X.46).7029N:<-"=?X/4 ` @H-F":<=<LK/[email protected] L;=QL;GM3TB?X 5 J.@M@H-":<GM/ e Ca-"=<GMFN02/D78,NVNB<gGMFN0@E?LNF"GM-":QV6%'8X+[ K];n<[;]K];Kp<)

[ b ) ^ ).(6).E?-"_02=<3 X ^ ).(6).E?-"_02=<3 XKco).(6)J.B<BKXT02/.3D(*).(,.-/ X4E?-@HRkg =<UT0/.GMl"-"3 O GHF"=<LKgiS L;=<LKGH3.B-"/.-"=?02:<-"3GH/10 602F"J.J.Cag e =<F;X 5D+J.@H@M-":<GH/ e Ca-"=<GMFN02/578,NV;B2GMFN02@E?LNF"GM-":AV6% - X+\;YK\an<ZKYKY;YKp<)

[;K)46)+(*) ^ 0<dNGM3.B2LK/10/.387)NS)NS*B?0/.UTX+78,NVNB?)*[email protected]];X8\;K\;Zn<[;]K ;p<)

[ K)46)8(*) ^ 0<dNGH3.B<LK/ X+~6);co)8(, 02/ X8(*)8(,.-/X802/.31E.)8NJ./T3 X84-d+) O LN3 )*78,NV;) , '8X8\ b [n<[K];]K[Kp<)

ZKf

Page 29: Equilibrium and Stability Properties of Self-Organized Electron ...

*# %aITGMU8)+[( =<L;B<B6B<-F":<GHLK/L;R*02/-@H-"F:<=<L;/B<m.GH=?02@+:<LK=<L;GM3B<,.L?_*GH/.U:<,T-m8LK@HLKGH3 02@+-"@M-"F":<=<LK/h.LK_ *,+ n p )- +

0/.3mTL;@ML;GM3 0@+FLNLK=<3TGM/ 0:<-B2V;B2:<-Cn# p _*GH:<, )- 1*mTL;GM/N:<GM/.UaGH/;:<La:<,.-m 02U;-N)ITGMU8)+Z78@HLK:<B6LKR*n<0;p /.LK=<CD02@HGHl-"3F"VNF@HLK:<=<L;/Rk=<-yNJ.-"/.F"V n "! p*02/.35nkW p /.L;=<C0@MGHl-"3-"@H-F":<=<LK/

0/.UKJ.@Q02=*h.L?_sdK-"@HL;F"GH:AV n"! p d;-=<B2J.B*:<,.-/.L;=<C0@MGHl-"3CaGH/.LK=*=?03.GMJ.B #6! RHL;=*:<,.-F"@MLNF29N_*GHB<-"gi=<L;:?02:<GM/TU-"@M-"F":<=<LK/hTL?_jGH/10:<-/NJ.LKJTB ` E?Ss-yNJ.GH@MGHW.=<GHJ.C)S*,.-F2,.L;GMF"-LKR B2V;B2:<-Cm 0=?02Ca-:<-"=<B*F"L;=<=Q-"B<m8LK/.3.B6:<LT|

j[ ;X 47s[ Y;ZKX sYKX+02/.3 "! s[ Y;[K)

ITGMU8)+\78@HLK:<B6LKR*n<0;p /.LK=<CD02@HGHl-"3F"VNF@HLK:<=<L;/Rk=<-yNJ.-"/.F"V n"! p*02/.35nkW p /.L;=<C0@MGHl-"3-"@H-F":<=<LK/0/.UKJ.@Q02=*h.L?_sdK-"@HL;F"GH:AV n "! p d;-=<B2J.B*:<,.-/.L;=<C0@MGHl-"3CaGH/.LK=*=?03.GMJ.B # "! RkL;=*:<,.-F"@MLNF29N_*GHB<-"gi=<L;:?02:<GM/TU-"@M-"F":<=<LK/hTL?_jGH/102/GM/N:<-"/.B<- ` E?Ss-yNJ.GH@MGHW.=<GMJTC) S6,.-F2,.LKGHF"-LKR B<VNB<:<-"Cm 0=?02Ca-:<-"=<B*F"L;=<=Q-"B<m8LK/.3.B6:<LT|

j[;YKY;YKX 47s[ YKYKY;[KX sYKX+02/.3 "! s[ Y;[K)

ITGMU8) b 78@HLK:<BLKRn<0Kp8/.LK=<CD02@HGMl"-"3=<-;[email protected]"/.Rk=<-"y;J.-"/.F"V4- n "! p02/T3nkW pT/.L;=<C0@MGHl-"3GHC0UKGH/ 02=<V-"GMU;-/TRH=<-"yNJ.-"/.FVPQC n"! p d;-=<B2J.B*:<,.-m8LK@HLKGH3 02@+CaL;3.-/NJ.CW8-"= RkLK=0:<-/NJ.L;J.B ` EKS_*GH:<,:<,.-BK02Ca-F2,.LKGHF"-L;R B<VNB<:<-"Ctm 02=?0C-":<-"=<B02B6B<,.L?_6/GM/5ITGMU8)8Z;)

ITGMU8)N78@HLK:LKRT:<,.-*/.LK=<CD02@HGMl"-"3GHCD02UKGH/ 02=<V-"GMUK-"/.Rk=<-"yNJ.-/.F"VPAC n "! pTdK-"=<B<J.B:<,T-3.GHC-"/.B<GHLK/.@H-"B<Bm 0=?02Ca-:<-"=

"!

n"! p6 RkLK=*B2-d;-=?0@[email protected] :<,.-m8LK@HLKGH3 02@+CaL;3.-CJ.CW8-= )*~-"=<-;X:<,.-F,.LKGHF-L;R B<VNB<:<-"Cm02=?02Ca-":<-=<B6FL;=<=<-B2mTL;/.3.B*:<L8| jY;X "! s[ YK[;X+02/.34 8 [jY Y;\KZ(

)

ITGMU8) 78@HLK:<BLKRn<0Kp8/.LK=<CD02@HGMl"-"3=<-;[email protected]"/.Rk=<-"y;J.-"/.F"V4- n"! p02/T3nkW pT/.L;=<C0@MGHl-"3GHC0UKGH/ 02=<V-"GMU;-/TRH=<-"yNJ.-"/.FVPQC n "! p d;-=<B2J.B*:<,.-m8LK@HLKGH3 02@+CaL;3.-/NJ.CW8-"= RkLK=0/GM/N:<-"/.B<- ` E?S_*GH:<,:<,.-BK02Ca-F2,.LKGHF"-L;R B<VNB<:<-"Ctm 02=?0C-":<-"=<B02B6B<,.L?_6/GM/5ITGMU8)8\;)

ZK

Page 30: Equilibrium and Stability Properties of Self-Organized Electron ...

29

Fig. 1

( )φρ,ρer

φer

2br

1br0

φφφeV r

( )φρ,ρer

φer

2br

1br0

φφφeV r

Page 31: Equilibrium and Stability Properties of Self-Organized Electron ...

30

Fig. 2

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

-1.6

-1.2

-0.8

-0.4

0.010

4 Ω s /ω

p(r b1

)

(a)

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

0.0

0.4

0.8

1.2

1.6

102 ω

b /ω

p(r b1

)

(b)

Page 32: Equilibrium and Stability Properties of Self-Organized Electron ...

31

Fig. 3

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

-3.0

-2.0

-1.0

0.010

4 Ω s /ω

p(r b1

)

(a)

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

0.0

0.4

0.8

1.2

103 ω

b /ω

p(r b1

)

(b)

Page 33: Equilibrium and Stability Properties of Self-Organized Electron ...

32

Fig. 4

0 2 4 6 8 10l

0.07

0.08

0.09

0.10

0.11

0.12R

e ω

/ωp(

r b1)

(a)

0 2 4 6 8 10l

-0.006

-0.003

0.000

0.003

0.006

Im ω

/ωp(

r b1)

(b)

Page 34: Equilibrium and Stability Properties of Self-Organized Electron ...

33

Fig. 5

0.0 0.5 1.0 1.5 2.0 2.5α

0.000

0.002

0.004

0.006

0.008

Im ω

/ωp(

r b1)

l=5l=6

l=7l=8

Page 35: Equilibrium and Stability Properties of Self-Organized Electron ...

34

Fig. 6

0 2 4 6 8 10l

0.0

0.2

0.4

0.6R

e ω

/ωp(

r b1)

(a)

0 2 4 6 8 10l

-0.03

-0.02

-0.01

0.00

0.01

Im ω

/ωp(

r b1)

(b)